{"id":"f61def28-bc5c-4877-a663-56d436791757","arxiv_id":"2506.07815","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Positive proportion of central L-values L(1/2,χ_c) are nonzero for all fixed orders ℓ≥3 in the Kummer function field setting.","lead":"A new proof shows that for every fixed character order ℓ≥3, a positive proportion of L-functions attached to order-ℓ characters over F_q[t] do not vanish at their central point. The result breaks the v=1 support barrier in the one-level density method and gives explicit nonvanishing proportions, such as 1/6 for ℓ=3.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The v>1 support in Theorem 1.1 rests entirely on Theorem 4.1/4.4 for the generating series ψ^(i), but §4 defines ψ with a different denominator than (4.1)-(4.2) use, and Theorem 4.1 is only sketched via imports. This is the least secure load-bearing point.","rationale":"The reader's weakest assumption pointed to the imported generating-series bounds from [Hof92], [Pat07], and [DFL22]; I agree that this is the right area. However, the concern is sharper and more concrete: Section 4 contains an internal inconsistency in the definition of ψ, and Theorem 4.1 is not proved in the paper for general ℓ, only sketched with references to the cubic case. Since the positive proportion depends on support v>1, and the margins above v=1 in Theorem 1.1 are only O(1/ℓ) for large ℓ, the argument is sensitive to the exact form of the convexity and residue bounds. This is a genuine conditional-acceptance issue. I do not think it overturns the result: the structural framework is coherent, the large sieve and Poisson-summation steps are present, and the paper itself notes that even a convexity-only treatment would yield positive proportion in several regimes, so the concern is about the completeness of the proof and the exact constants, not about a visible fatal contradiction. The verdict should remain CONDITIONAL, requiring the authors to correct the definition of ψ, give a self-contained proof (or precise general-ℓ citations) for Theorem 4.1, and state explicitly that the Lindelöf bound used in Section 7.1 is the known Weil/RH bound over function fields. For these reasons the reader's conditional verdict is the right one, and no change of verdict is needed.","tokens_in":49198,"tokens_out":43248,"duration_ms":462464,"concrete_test":"For ℓ=4 and q=5, compute C(1,k) from (4.3) for k≤8 by direct enumeration of G_4(1,F), substitute into (4.4) with B=3, and form the rational function in (4.2). Compare its power series coefficients with the definition in Section 4. If the coefficients disagree, the manuscript's ψ is not consistently defined, and the proof of Theorem 4.1 must be redone after correcting the denominator. Independently re-derive the two cases in the proof of Theorem 4.1 (i≡j and i≠j mod ℓ) for general ℓ; if either case requires an assumption not stated, recompute the Type I bounds in Section 9 and the support thresholds in Theorem 1.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every bound that pushes the Fourier support past (−1,1) in Sections 6–10 reduces to controlling averages of ψ^(i)(hV^j,z) and of the residue of ψ at u^ℓ=q^{−ℓ−1}; this is the mechanism behind Theorem 1.1. Two features make this the least secure part of the paper. First, the definition in Section 4, ψ^(i)(r,u) := (1−q^ℓ u^ℓ)^{−1} Σ ... , has a pole at u^ℓ=q^{−ℓ}, while equation (4.2) and the surrounding text declare that ψ has poles at u^ℓ=q^{−ℓ−1}, with denominator 1−q^{ℓ+1}u^ℓ. The functional equation (4.1) and the approximate functional equation (Theorem 4.4) are written for the latter object, so as printed the object whose convexity is proved is not the object defined. Second, Theorem 4.1 is only sketched: the functional equation is imported from [Hof92, Pat07] and the case analysis (i≡j mod ℓ, linear dependence of (a1,a2) and (b2,b1), equations (4.5) and the surrounding argument) is delegated to [DFL22], which is written for ℓ=3. If any of these steps fails for general ℓ, or if the true convexity exponent is worse by a term of size n, the support v=1+O(1/ℓ) in Theorem 1.1 would not follow. Section 7.1 additionally invokes the 'Lindelöf hypothesis' without saying it is the known function-field RH; this is a clarity issue, but the generating-series issue is the substantive one.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the family F_ℓ(d) of order-ℓ Dirichlet characters over F_q[t] in the Kummer setting q≡1 mod 2ℓ with ℓ∤d. Theorem 1.1 computes the one-level density of low-lying zeros for this family, with test functions whose Fourier transform is supported in (-v,v) for a piecewise-defined v>1; Corollary 1.2 converts this into a positive proportion of nonvanishing central values L(1/2,χ_c), for example at least 1/6 for ℓ=3 and 3/26 for ℓ=4. The proof uses the explicit formula, Poisson summation, Vaughan's identity, a combinatorial decomposition of generating series of Gauss sums (Theorem 5.3), Lindelöf-on-average bounds for these series (Section 6), large-sieve inequalities (Theorems 1.4 and 1.5), and a Type I/Type II analysis. The paper also proves Theorem 1.6, a cancellation bound for averaged shifted Gauss sums at prime arguments.","tokens_in":49578,"tokens_out":35409,"duration_ms":406708,"significance":"If the analytic ingredients are valid, this is a substantial advance: it gives the first positive-proportion nonvanishing result for fixed-order characters of arbitrary order ℓ in the function-field Kummer setting, improving on the infinitely-many result of Ellenberg–Li–Shusterman. The proof is highly structured and includes useful standalone tools, in particular large-sieve inequalities for order-ℓ characters and a uniform Gauss-sum cancellation result. The main risk is the imported analytic theory for the generating series ψ^(i): the paper relies on published work of Hoffstein, Patterson, and the authors' own ℓ=3 paper, and the exact form of the required statements and exponents is not fully pinned down in the present text. These issues are fixable but load-bearing.","major_comments":[{"comment":"The function defined at the start of §4 is ψ(i)(r,u)=(1−q^ℓ u^ℓ)^{-1}Σ..., while Eq. (4.1), Eq. (4.2), and Theorem 4.4 concern an object with denominator 1−q^{ℓ+1}u^ℓ and poles at u^ℓ=q^{−ℓ−1}. The manuscript never explains the cancellation that reconciles these two objects; without it, the convexity bound in Theorem 4.1 on q^{−3/2}≤|u|≤q^{−1/2}, away only from u^ℓ=q^{−ℓ−1}, would fail at the pole u^ℓ=q^{−ℓ} of the displayed definition. The proof of Theorem 4.1 is also only a sketch: the case analysis for i≠j is delegated to [DFL22], which treats ℓ=3, and the displayed relation for ψ(j)(r,s) has ψ(j) on both sides (the second term should be ψ(i)(r,2−s)). Since the v>1 support in Theorem 1.1 is inherited from the Lindelöf-on-average bounds of Section 6, which use Theorems 4.1 and 4.4, this point must be stated precisely and proved completely.","section":"§4, Eqs. (4.1)–(4.4), Theorems 4.1 and 4.4"},{"comment":"Theorem 1.4 is stated only for the character (M/N)_ℓ, but the Type II application in (8.2)–(8.3) requires the same bound for (α/c)^2_ℓ, and Lemma 12.2 requires it for (M_k/N)^k_ℓ for each 1≤k≤ℓ−1. For even ℓ, χ^2 has order ℓ/2, so the statement as written does not apply; this matters already for ℓ=4. The proof of Theorem 1.4 appears to go through unchanged for any fixed k, so please state and prove the generalized large sieve for all nontrivial powers, or supply the additional quadratic/cubic large-sieve input needed for even ℓ.","section":"§8, Eq. (8.3); §12, Lemma 12.2 and Theorem 1.4"},{"comment":"The denominator product in Theorem 5.3 is not consistent with the factors derived in the proof. In the step-by-step derivation, a prime π|r_{ℓ−1} contributes a factor with exponent 1 and no (−1/π)_ℓ symbol, while a prime π|aE contributes exponent ℓ−1; the displayed product over π|aEr_1...r_{ℓ−1} with exponent ν_π(aEr_1...r_{ℓ−2}r_{ℓ−1})+1 gives exponent 2 for π|r_{ℓ−1} and, typically, 2 or 3 for π|aE. Please correct the formula, or state explicitly that only the O(1) size of these factors is used in the later bounds. Since Theorem 5.3 is the pivot for Corollary 5.4, Proposition 6.3, and the Type I bounds, the exact statement must be reliable.","section":"§5, Theorem 5.3"}],"minor_comments":[{"comment":"The phrase 'using the Lindelöf hypothesis' for the bound Σ_{c∈H_d} χ_c(f)≪q^{d/2+εd} should be replaced by a precise reference to the relevant Weil/Riemann-hypothesis bound in function fields; as written, it sounds like an unproved assumption, even though the paper's results are meant to be unconditional.","section":"§7.1"},{"comment":"In the displayed formula for ψ(j)(r,s), the second term on the right-hand side repeats ψ(j)(r,2−s); it should be ψ(i)(r,2−s), otherwise the displayed identity is tautological.","section":"§4, proof of Theorem 4.1"},{"comment":"The range '1≤j≤n−2' should be '1≤j≤ℓ−2'; the letter n is not defined in that lemma.","section":"§5, Lemma 5.1(5.3)"},{"comment":"After Eq. (7.2), the notation F_ℓ(g) should be F_ℓ(d).","section":"§10.1"},{"comment":"The comparison statement in Remark 6.5 says 'for ℓ≥8' but the condition in Corollary 5.4 at σ=1+1/ℓ appears to switch cases at ℓ>8; please check whether ℓ=8 belongs to the first or second case.","section":"§6, Remark 6.5"}],"recommendation":"major_revision","confidential_remarks":"The paper's overall architecture is convincing and the result is important if the analytic inputs hold. My main concerns are the §4 definition-versus-functional-equation mismatch, the incomplete proof of the convexity/approximate-functional-equation input for general ℓ, and the fact that the large sieve is stated only for the first power of the ℓ-th residue symbol although the applications need higher powers. All of these appear fixable with a careful revision, so I am not recommending rejection; I would need to see the corrected statements and proofs before supporting acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is the first positive-proportion nonvanishing result for L(1/2, χ_c) where χ_c is an order-ℓ character over F_q[t] for arbitrary fixed ℓ≥3. Previous positive proportions only covered ℓ=2,3,4; general ℓ only had infinitely many nonvanishing twists (ELS20). The authors also get support v>1 in the one-level density, a new large sieve for order-ℓ characters, and a cancellation bound for shifted Gauss sums at primes. The main theorem has explicit proportions (1/6 for ℓ=3, etc.). All of that is real.\n\nThe paper is long and carefully structured: approximate functional equation, Vaughan's identity, Type I/II sums, large sieve. The proofs are mostly there, and the external inputs are standard or published. I believe the central argument is sound at the structural level.\n\nThe soft spots, in order of real softness. First, there is a genuine mismatch in Section 4: ψ^(i)(r,u) is defined with denominator (1 - q^ℓ u^ℓ), but the functional equation and equation (4.2) use (1 - q^{ℓ+1} u^ℓ). Those are different objects; the first has a pole at u^ℓ = q^{-ℓ}, the second at q^{-ℓ-1}. This is probably a typo in the definition, but it is load-bearing: Theorem 4.1, the approximate functional equation, and everything after depends on the q^{ℓ+1} version. A referee must ask the authors to fix it and confirm the residue statements. Second, Theorem 4.1 is only sketched, with the messy case analysis imported from [DFL22], which is written for ℓ=3. The generalization to all ℓ is plausible but not fully shown in this preprint. Third, §7.1 invokes the Lindelöf hypothesis without saying it is the known function-field theorem; that's a clarity issue.\n\nNone of these, on the evidence here, overturns the result. But they are exactly why the right verdict is conditional, not unconditional. The paper deserves a serious referee: it is important, the techniques are novel, and the gaps are addressable. I would recommend sending it out with a request to fix the generating-series definition and to either prove Theorem 4.1 in the generality claimed or give a precise pointer. If those checks pass, the result stands.","headline":"Genuinely new positive-proportion nonvanishing for arbitrary fixed order ℓ in function fields; substantial but needs referee attention to a generating-series definition mismatch.","tokens_in":50128,"tokens_out":3557,"would_cite":true,"duration_ms":40919,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M38","11R16","11R58"],"pacs":[],"model":"deepseek-v4-flash","headline":"Positive share of order-$\\ell$ $L$-functions avoid vanishing at the central point","keywords":["function fields","non-vanishing","one-level density","low-lying zeros","fixed order characters","Gauss sums","large sieve","L-functions"],"falsifier":"A direct check would be to run the large-sieve inequality of Theorem 1.4 on an explicit sequence of coefficients $\\lambda(N)$: a choice of supports $m,n$ that violates the stated $q^{2(m+n)/3}$ term would collapse the Type II bound. A complementary check is to enumerate $F_\\ell(d)$ for a fixed small $\\ell$ and growing $d$ and compare the nonvanishing proportion against the Corollary 1.2 fraction, though only a persistent violation as $d$ grows would be decisive.","tokens_in":48986,"feed_emoji":"🔢","tokens_out":10826,"duration_ms":100922,"temperature":0.7,"pith_summary":"The paper proves that, for every fixed order $\\ell\\ge 3$ with $q\\equiv 1 \\pmod{2\\ell}$ and $\\ell\\nmid \\deg(c)$, a positive proportion of the $L$-functions attached to the $\\ell$-th residue characters over $\\mathbb{F}_q[t]$ do not vanish at $s=1/2$ as $\\deg(c)\\to\\infty$. This upgrades the previous infinitude statement for general $\\ell$ to an explicit positive fraction of the family $F_\\ell(d)$. The proof uses a one-level density computation for the low-lying zeros, pushing the Fourier support of the test function beyond the interval $(-1,1)$, which is exactly what forces a positive proportion of nonvanishing for unitary families. The stated fractions are $1/6$ for cubic characters, $3/26$ for quartic characters, and smaller explicit positive fractions for every $\\ell\\ge 5$.","feed_headline":"Positive share of order-$\\ell$ $L$-functions avoid vanishing at central point","feed_subtitle":"One-level density past the $(-1,1)$ barrier gives explicit nonvanishing fractions, from $1/6$ for cubic to small positives for all $\\ell$.","key_machinery":"The load-bearing object is the generating series $\\psi^{(i)}(r,u)=(1-q^\\ell u^\\ell)^{-1}\\sum_{\\deg F\\equiv i\\pmod{\\ell}} G_\\ell(r,F)u^{\\deg F}$, which packages shifted order-$\\ell$ Gauss sums by degree class. The paper combines the rationality and functional equation of these series with a new approximate functional equation for their numerators, then controls their average size with large-sieve inequalities for order-$\\ell$ characters and a Vaughan-identity decomposition of the sums over primes. Beating the pointwise convexity bound on these averages is what pushes the one-level density support past the $(-1,1)$ barrier.","core_discovery":"The central claim is Theorem 1.1: for the family $F_\\ell(d)$ of order-$\\ell$ residue characters with square-free modulus of degree $d$, the one-level density of zeros equals $\\widehat{\\varphi}(0)+O(1/d)$ for even Schwartz test functions whose Fourier transform is supported in $(-v,v)$, with $v$ strictly larger than $1$ and given piecewise: $v=6/5$ for $\\ell=3$, $v=26/23$ for $\\ell=4$, and slightly larger than $1$ for $\\ell\\ge 5$. Because the symmetry type is unitary, the condition $v>1$ is precisely what lets the density force a zero at the central point, and Corollary 1.2 converts this into the nonvanishing proportions. As a by-product, Theorem 1.6 establishes cancellation in averages of shifted order-$\\ell$ Gauss sums $G_\\ell(R,\\pi)$ over primes $\\pi$, uniformly in the shift $R$, which the paper presents as a step toward equidistribution of their angles.","pith_inferences":["Looking beyond the paper, any strengthening of the Lindelöf-on-average bounds for $\\psi^{(i)}$ would enlarge the allowed support $v$ and thereby raise the nonvanishing fractions; the authors explicitly note they did not optimize small $\\ell$ cases to keep the paper shorter.","Theorem 1.6 stops short of full equidistribution of the angles of $G_\\ell(R,\\pi)$ because only a restricted class of generating averages is bounded; extending the argument to higher moments would supply Weyl-criterion equidistribution with uniformity in $R$.","Since the Corollary 1.2 fractions decay like $O(1/\\ell)$ as $\\ell$ grows, the method does not address Chowla-type nonvanishing of every character; the paper itself remarks that the proportion approaches zero as $\\ell\\to\\infty$.","The same template—Vaughan's identity, Poisson summation, and the order-$\\ell$ large sieve—should apply to other thin families of fixed-order characters in function fields, such as non-Kummer settings where the base field lacks the $\\ell$-th roots of unity."],"forward_implications":["For every $\\ell\\ge 3$ there are positive proportions of nonvanishing central values in the family $F_\\ell(d)$, with the explicit fractions $1/6$ for $\\ell=3$, $3/26$ for $\\ell=4$, and $2(\\ell-2)/(2\\ell^2+\\ell-2)$, $2(\\ell-2)/(3\\ell^2-7\\ell-2)$, or $6(\\ell-2)/(9\\ell^2-25\\ell-6)$ in the stated ranges of $\\ell$.","The one-level density matches the unitary prediction $\\widehat{\\varphi}(0)$ up to $O(1/d)$, for Fourier support reaching $v>1$ in every case considered.","Averaged shifted order-$\\ell$ Gauss sums over primes have cancellation bounds that are uniform in the shift $R$, a feature not available for general $\\ell$ in the number-field setting.","The unconditional large-sieve inequalities for order-$\\ell$ characters are proved as standalone theorems and can be used in other families."],"supporting_citations":[{"why":"Establishes rationality and the functional equation for the generating series $\\psi^{(i)}$ in the function-field setting.","marker":"[Hof92]"},{"why":"Supplies the order-$\\ell$ convexity and residue bounds imported as Theorem 4.1.","marker":"[Pat07]"},{"why":"Gives the cubic-function-field identities and bounds that the present paper generalizes to all $\\ell$.","marker":"[DFL22]"},{"why":"Provides the large-sieve inequality for fixed-order characters that Theorems 1.4 and 1.5 refine.","marker":"[BGL14]"},{"why":"The earlier infinitude nonvanishing theorem for order-$\\ell$ characters that this paper improves to a positive proportion.","marker":"[ELS20]"},{"why":"Introduces the Vaughan-identity decomposition of prime sums of Gauss sums used in Section 7.","marker":"[HBP79]"},{"why":"Contributes the large-sieve ideas for character sums that underpin the Type II bounds.","marker":"[HB95]"},{"why":"Provides the cubic Gauss sum estimates whose pattern the present Type I/Type II analysis follows.","marker":"[HB00]"},{"why":"Demonstrates the approximate-functional-equation and residue technique for bounding averages of the generating series.","marker":"[DdFDS]"}],"fun_headline_variants":["Positive proportion of order-l L-values nonzero","Busting the (-1,1) barrier: nonvanishing L-values for order-l chars","Cubic L-functions: at least 1/6 nonzero at central point","Nonvanishing order-l L-values over function fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the imported bounds for the generating series of shifted Gauss sums are exactly as stated in Theorem 4.1; if those bounds were weaker, the error terms would not be small enough to push the one-level density support beyond 1, and the positive proportion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Positive proportion of order-l L-values nonzero","Busting the (-1,1) barrier: nonvanishing L-values for order-l chars","Cubic L-functions: at least 1/6 nonzero at central point","Nonvanishing order-l L-values over function fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001316,"raw_usage":{"total_tokens":5422,"prompt_tokens":1067,"completion_tokens":4355,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":683,"completion_tokens_details":{"reasoning_tokens":4278}},"tokens_in":683,"tokens_out":4355,"duration_ms":35738,"temperature":1.0,"reasoning_tokens":4278,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:26:04.643624+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check would be to run the large-sieve inequality of Theorem 1.4 on an explicit sequence of coefficients $\\lambda(N)$: a choice of supports $m,n$ that violates the stated $q^{2(m+n)/3}$ term would collapse the Type II bound. A complementary check is to enumerate $F_\\ell(d)$ for a fixed small $\\ell$ and growing $d$ and compare the nonvanishing proportion against the Corollary 1.2 fraction, though only a persistent violation as $d$ grows would be decisive.","supporting_citations":[],"review_version":1}